What is another way we can get to the 5th Station from Shinjuku in the early AM?

by Chelsea
(Gold Coast, Australia)

Hello,
We are looking to climb Mt Fuji on Tuesday 5th July and will be leaving from our hotel in Shinjuku. Looking at the bus timetables the earliest bus from Shinjuku to 5th Station leaves at 6:40am, however would like to commence our ascent at around 6-6:30am.

Due to time restraints we are looking at completing the climb and descend in 1 day. All Fit hikers. What is another way we can get to 5th Station from Shinjuku in the early AM. Would getting a taxi the whole way be the easiest? Or train?

Many Thanks
Chelsea